Projective classification of rational \(\mathbb {C}\mathbf {P}^{1}\)-mappings


We study the orbits of various \(\mathbf {SL}_{2}\left( \mathbb {C}\right) \)-actions on the spaces of rational \(\mathbb {C}\mathbf {P}^{1}\)-mappings. The fields of rational differential invariants and the corresponding ordinary differential equations that describe orbits are found.

This is a preview of subscription content, log in to check access.


  1. 1.

    Bibikov, P., Lychagin, V.: Projective classification of binary and ternary forms. J. Geom. Phys. 61(10), 1914–1927 (2011)

    Google Scholar 

  2. 2.

    Bibikov, P., Lychagin, V.: On differential invariants of actions of semisimple Lie groups. J. Geom. Phys. 85, 99–105 (2014)

    Google Scholar 

  3. 3.

    Bogomolov, F., Petrov, T.: Algebraic curves and one-dimensional fields. Courant Lect. Math. 8, 214 (2002)

    Google Scholar 

  4. 4.

    Konovenko, N., Lychagin, V.: On projective classification of algebraic curves. Math. Bull. Shevchenko Sci. Soc. 10, 51–64 (2013).

    Google Scholar 

  5. 5.

    Krasilshchik, I.S., Lychagin, V.V., Vinogradov, A.M.: Geometry of jet spaces and nonlinear partial differential equations. Advanced Studies in Contemporary Mathematics, vol. 1. Gordon and Breach Science Publishers, New York, xx+441 pp (1986)

  6. 6.

    Kruglikov, B., Lychagin, V.: Geometry of differential equations. In: Hanbook of Global Analysis, pp. 725–772 (2008)

  7. 7.

    Konovenko, N.: Differential Invariants and \(\mathfrak{sl}_{2}\)-Geometries, p. 188. Naukova Dumka, Kiev (2013). (in Russian)

    Google Scholar 

  8. 8.

    Kumpera, A.: Invariants differentiels d’un pseudogroupe de Lie, Part 1. J. Differ. Geom. 10(2), 289–345 (1975)

    Google Scholar 

  9. 9.

    Kumpera, A.: Invariants differentiels d’un pseudogroupe de Lie, Part 2. J. Differ. Geom. 10(3), 347–416 (1975)

    Google Scholar 

  10. 10.

    Kruglikov, B., Lychagin, V.: Global Lie–Tresse theorem. Sel. Math. 22(3), 1357–1411 (2016)

    Google Scholar 

  11. 11.

    Olver, P.: Classical Invariant Theory. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  12. 12.

    Olver, P.: Differential invariant algebras. Comtemp. Math. 549, 95–121 (2011)

    Google Scholar 

  13. 13.

    Ovsiannikov, L.: Group Analysis of Differential Equations, pp. 487–489. Nauka, Moscow (1978). (Engl. transl. Academic Press New York (1982))

    Google Scholar 

  14. 14.

    Rosenlicht, M.: A remark on quotient spaces. An. Acad. Brasil. Ci. 35, 487–489 (1963)

    Google Scholar 

  15. 15.

    Tresse, A.: Sur les invariants differentiels des groupes continus de transformations. Acta Math. 18, 1–88 (1894)

    Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Konovenko Nadiia.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nadiia, K., Valentin, L. Projective classification of rational \(\mathbb {C}\mathbf {P}^{1}\)-mappings. Anal.Math.Phys. 9, 1865–1876 (2019).

Download citation